# Question 2.5.3: An Example of Subsystems: Two Coupled Tanks Consider two bri...

An Example of Subsystems: Two Coupled Tanks

Consider two brine tanks each containing 500 L (liters) of brine connected as shown in Figure 2.5.1. At any time t, the first and the second tank contain $x_1(t)$ and $x_2(t)$ kg of salt, respectively. The brine concentration in each tank is kept uniform by continuous stirring. Brine containing r kg of salt per liter is entering the first tank at a rate of 15 L/min, and fresh water is entering the second tank at a rate of 5 L/min. The incoming brine density r(t) can be changed to regulate the process, so r(t) is an input variable.

Brine is pumped from the first tank to the second one at a rate of 60 L/min and from the second tank to the first one at a rate of 45 L/min. Brine is discharged from the second tank at a rate of 20 L/min.

a. Obtain the differential equations, in terms of $x_1$ and $x_2,$ that describe the salt content in each tank as a function of time.
b. Obtain the transfer functions $X_1(s)$/R(s) and $X_2(s)$/R(s).
c. Suppose that r(t) = 0.2 kg/L. Determine the steady-state values of $x_1$ and $x_2$, and estimate how long it will take to reach steady state.

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